Question

In the figure given below the value of x and y would be ______ .

• A. 10 and 15
• B. 15 and 10
• C. 06 and 12
• D. 12 and 06

Answer 1

The Correct Option is B

To solve for the values of \(x\) and \(y\) in the given triangle, we need to utilize the fact that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Given the sides are \(4x\), \(2x + 3y\), and \(5y + 10\), we apply the triangle inequality theorem:

1. \(4x + (2x + 3y) > 5y + 10\) 
2. \(4x + (5y + 10) > 2x + 3y\)
3. \((2x + 3y) + (5y + 10) > 4x\)

We will address these inequalities one by one:

1. Expanding: \(4x + 2x + 3y > 5y + 10\) simplifies to: \(6x + 3y > 5y + 10\).
2. Rearranging gives: \(6x > 2y + 10\).
3. Further simplifying: \(6x - 2y > 10\). (Equation 1)
4. Expanding: \(4x + 5y + 10 > 2x + 3y\) simplifies to: \(4x + 5y + 10 > 2x + 3y\).
5. Rearranging gives: \(2x + 2y > -10\).
6. Simplifying: \(2x + 2y > -10\). (Equation 2)
7. Expanding: \(2x + 3y + 5y + 10 > 4x\) simplifies to: \(2x + 8y + 10 > 4x\).
8. Rearranging gives: \(8y > 2x - 10\).
9. Simplifying: \(8y - 2x > -10\). (Equation 3)

To solve these equations, we match the given options against equations derived. By testing \((x, y) = (15, 10)\):

1. Substitute \(x = 15, y = 10\) in Equation 1: \(6(15) - 2(10) > 10\) simplifies to \(90 - 20 = 70 > 10\), which holds true.
2. Substitute in Equation 2: \(2(15) + 2(10) > -10\) simplifies to \(30 + 20 = 50 > -10\), which holds true.
3. Substitute in Equation 3: \(8(10) - 2(15) > -10\) simplifies to \(80 - 30 = 50 > -10\), which holds true.

Thus, the correct values satisfying these inequalities are \(x = 15, y = 10\).